Configuration mixing of angular-momentum and particle-number projected triaxial Hartree-Fock-Bogoliubov states using the Skyrme energy density functional

We present a method based on mean-field states generated by triaxial quadrupole constraints that are projected on particle number and angular momentum and mixed by the generator coordinate method on the quadrupole moment. This method is equivalent to a seven-dimensional GCM calculation, mixing all five degrees of freedom of the quadrupole operator and the gauge angles for protons and neutrons. A first application to ${}^{24}\mathrm{Mg}$ permits a detailed analysis of the effects of triaxial deformations and of $K$ mixing.

Figure 1

Lower panel: Nilsson diagram of the eigenvalues of the proton single-particle Hamiltonian for axial shapes. Solid lines denote levels with positive parity, dotted lines levels with negative parity, and the red dashed line the Fermi energy. Upper panel: Particle-number projected (solid line) and particle-number and angular-momentum $J=0$ projected (dotted line) axial deformation energy curves.

Figure 2

Triaxial quadrupole energy maps obtained by projecting mean-field configurations on $N$ and $Z$ only (top) and also either on $J=0$ (middle) or on $J=2$ and $K=0$. In the latter case, the $z$ axis is chosen as the symmetry axis. Filled circles denote minima; the ⊗ symbols denote saddle points. All energies are normalized with respect to the energy of the particle-number projected spherical state.

Figure 3

Excitation spectra and selected $B(E2)$ values after decomposition of the mean-field state that gives the minimum of the $J=0$ projected energy surface into angular-momentum projected components $|JMKq\rangle$ (left) and after $K$ mixing into $|JM\kappa q\rangle$ (right) for three different orientations of the same triaxial state in the intrinsic frame.

Figure 4

Decomposition of the mean-field state that gives the minimum of the $J=0$ projected energy surface into components of same angular momentum $J$, Eq. (19). The results obtained from the different possible orientations in the intrinsic frame are undistinguishable.

Figure 5

Decomposition of the mean-field state that gives the minimum of the $J=0$ projected energy surface into components of same $K$, Eq. (20), for three different orientations of the state in the intrinsic frame.

Figure 6

Excitation spectra and selected $B(E2)$ values after decomposition of the mean-field state that gives the axial oblate saddle of the $J=0$ projected energy surface into angular-momentum projected components $|JMKq\rangle$. The left panel provides the decomposition when choosing the $z$ axis as the symmetry axis, $\gamma ={180}^{°}$. The middle panel gives the decomposition into $K$ components when the symmetry axis is chosen perpendicular to the $z$ axis. The right panel shows the unique band resulting from $K$ mixing (see text).

Figure 7

Variation of the eigenvalues of the $J=0$ norm kernel among different bases.

Figure 8

Deformation energy curve projected on $N$ and $Z$ (black) only and projected also on $J=0$ (red) for three different values of γ, as indicated. The energies of the $J=0$ GCM ground state obtained from the axial, triaxial, and full bases as described in the text cannot be distinguished within the resolution of the plot and are represented by the same filled diamond plotted at arbitrary deformation. All energies are normalized to that of the spherical particle-number projected state.

Figure 9

Excitation spectra and $B(E2)$ values in $e^2$ fm${}^4$ obtained in three of the bases defined in the text compared to the available experimental data. In the four cases, the spectrum is subdivided into a ground-state band, a γ band, and additional low-lying states that do not necessarily form a band.